A differential system [A]:Y′=AY, with A∈Mat(n,kˉ)
is said to be in reduced form if A∈g(kˉ) where
g is the Lie algebra of the differential Galois group G of
[A]. In this article, we give a constructive criterion for a system to be in
reduced form. When G is reductive and unimodular, the system [A] is in
reduced form if and only if all of its invariants (rational solutions of
appropriate symmetric powers) have constant coefficients (instead of rational
functions). When G is non-reductive, we give a similar characterization via
the semi-invariants of G. In the reductive case, we propose a decision
procedure for putting the system into reduced form which, in turn, gives a
constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr